# The Klein View of Geometry

Fundamental Theorems

 In euclidean geometry, for two points A and A', there are elements of E(2) which map A to A'. A translation is one possibility. In our other geometries, a similar result holds. In the similarity and inversive geometries, this is easy to see. In hyperbolic geometry, it is a consequence of the Interchange Lemma. However, in euclidean geometry, if L = (A,B) and L' = (A',B') are lists of distinct points, then we can map L to L' if and only if |AB| =|A'B'|. In similarity geometry, we can map pairs to pairs. We shall call this The Fundamental Theorem of Similarity Geometry If L = (A,B) and L' = (A',B') are lists of distinct points of E, then there is a unique element of S+(2) which maps L to L'. We can go no further since similarity geometry as the property of angle. Thus we can map (A,B,C) to (A',B',C') if and only if the three angles defined by each list are equal. In inversive geometry, the situation is more interesting. We have The Fundamental Theorem of Inversive Geometry If L = (A,B,C) and L' = (A',B',C') are lists of distinct points of E+, then there is a unique element of I+(2) which maps L to L'. This result and its proof appear on the inversive group pages. In hyperbolic geometry, we have the property of hyperbolic distance, so the situation is similar to that in euclidean geometry. The other geometries we consider, affine and projective, have the property that two distinct points lie on a unique line. Also, the group elements map lines to lines, so we cannot hope to map three collinear points to three non-collinear points, or vice versa. With this restriction, we do have Fundamental Theorems for these geometries. The Fundamental Theorem of Affine Geometry If L = (A,B,C) and L' = (A',B',C') are lists of non-collinear points of E, then there is a unique element of A(2) which maps L to L'. A proof can be found in the affine group pages. Since affine transformations preserve ratio along a line, we cannot in general map three collinear points to another three collinear points. The Fundamental Theorem of Projective Geometry If L = (A,B,C,D) and L' = (A',B',C',D) are lists of points of RP2, each with no three collinear, then there is a unique element of P(2) which maps L to L'. A proof can be found in the projective group pages. Since projective transformations preserve the cross-ratio of four collinear p-points, we cannot in general map four collinear points of RP2 to another four collinear points.